On local antimagic chromatic numbers of circulant graphs join with null graphs or cycles

• Lau, G. C. ^[1] ; Premalatha, K. ^[3] ; Shiu, W. C. ^[2] ; Nalliah, M. ^[4]
  1. [1] Universiti Teknologi MARA

     Universiti Teknologi MARA

     Malasia

     
  2. [2] Chinese University of Hong Kong

     Chinese University of Hong Kong

     RAE de Hong Kong (China)

     
  3. [3] Sri Shakthi Institute of Engineering and Technology.
  4. [4] Vellore Institute of Technology.

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• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 42, Nº. 5, 2023, págs. 1307-1332
• Idioma: inglés
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• Resumen

  • An edge labeling of a graph G = (V,E) is said to be local antimagic if there is a bijection f : E → {1,..., |E|} such that for any pair of adjacent vertices x and y, f +(x) ≠ f +(y), where the induced vertex label is f +(x) = 𝜮 f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χla(G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. For a bipartite circulant graph G, it is known that χ(G)=2 but χla(G) ≥ 3. Moreover, χla(Cn ∨ K1)=3 (respectively 4) if n is even (respectively odd). Let G be a graph of order m ≥ 3. In [Affirmative solutions on local antimagic chromatic number, Graphs Combin., 36 (2020), 1337—1354], the authors proved that if m ≡ n (mod 2) with χla(G) = χ(G), m>n ≥ 4 and m ≥ n2/2, then χla(G∨On) = χla(G)+1. In this paper, we show that the conditions can be omitted in obtaining χla(G∨H) for some circulant graph G, and H is a null graph or a cycle. The local antimagic chromatic number of certain wheel related graphs are also obtained.